Contents

1. Background and motivation

2. General procedure

   1. terminology

      1. the notion of space

      2. the notion of path

      3. the notion of trajectory

      4. the notion of background field

      5. the notion of action functional

      6. the notion of section

      7. the notion of state

   2. the quantum propagation map

3. Examples

   1. the electromagnetically charged particle

   2. Dijkgraaf-Witten theory

   3. Yetter model

Background and motivation

The search is on for the abstract formalization of the process of quantization – the process that reads in a “classical field theory” for instance presented in form of a gauge theory or in form of a σ-model background field data – and spits out the corresponding quantum field theory.

While for various aspects and facets of this question there are well-developed formalisms – such as geometric quantization or deformation quantization? or BV theory – a full answer is certainly still missing, not the least because the full formalization of the question itself has still to be established.

Considerable progress on this formulation of the question has been achieved with the formalization and proof of the cobordism hypothesis in On the Classification of Topological Field Theories by Jacob Lurie. This at least indicates what the result of any full quantization procedure should be in that it clarifies what exactly a TQFT FQFT is: a morphism from the (∞,n)-category of cobordisms $Z:{\mathrm{Bord}}_n\to C$.

In On the Classification of Topological Field Theories Jacob Lurie only indicates the desire to find a similar formalization of “classical field theory” and a systematic procedure for turning the classical theory into the quantum theory.

These thoughts were further developed in TFT from compact Lie groups, but remain sketchy for the moment.

For the purpose of the present entry this indication of a quantizaton proposal by Lurie et al. mainly serves as a reference for the idea itself that a formalization of something interesting is to be sought here, and of the kind of abstract nonsense answer one hopes to find. We will however discuss a somewhat different-looking approach. It may well be related to the Lurie-et al proposal in the end, but for the time being we shall not concentrate on that relation.

Rather, the approach for a formalization of the quantization procedure that shall be discussed at this entry here is more in the spirit as indicated to some extent in the work of David Ben-Zvi et. al, which is described in some detail at geometric ∞-function theory. This is essentially based on the idea that a quantum field theory is obtained from a σ-model target space object $X$ by homming extended cobordism cospans ${\Sigma }_{\mathrm{in}}\to \Sigma \leftarrow {\Sigma }_{\mathrm{out}}$ into the target object and then pull-pushing geometric function objects through the resulting spans of configuration space objects $[{\Sigma }_{\mathrm{in}},X]\leftarrow [\Sigma ,X]\to [{\Sigma }_{\mathrm{out}},X]$.

The resulting pull-push operation is an example or a generalization of what John Baez discusses under the term groupoidification. David Ben-Zvi et al would speak of geometric function theory.

The main result of David Ben-Zvi et. al.’s work on this approach is that they point out that as soon as the geometric function object one uses satisfies the two fundamental theorems of geometric infinity-function theory, a considerable amount of rich structure that has in parts been known by itself gets unified into one coherent elegant story: the nature of partition functions (i.e. traces), of centers, of Hochschild (co)homology, Deligne-Kontsevich-statements, etc. all are understood by means of a suitable geometric function theory as induced from the underlying geometry of configuration space objects $[\Sigma ,X]$ as well as the loop space objects of $X$.

Here in this entry the aim is to discuss in detail aspects and models for the quantization procedure that is suggested by applying geometric function theory to σ-model backgrounds.

But more precisely, the premise of the discussion here is that a fundamental understanding of the situation is obtained after regarding seriously the fact that

1. background fields are not just encoded by the cohomology with coefficients in the target space object $X$, but by the corresponding differential cohomology;

2. and that the functorial notion of the FQFT that one wants to obtain suggests that these differential cocycles themselves are encoded functorially, as described at differential nonabelian cohomology.

This means that we shall take the classical structure $\nabla$ which the sought-after quantization procedure is supposed to send to an FQFT $Z_{\nabla }:{\mathrm{Bord}}_n\to C$ to be, essentially, a functor

from some path n-groupoid of $X$ to some coefficient object $A$.

Here the k-morphisms in ${\Pi }_n(X)$ are generated from the existing $k$-morphisms in $X$ as well as the $k$-dimensional smooth paths in $X$. The functor $\nabla$ on this encodes the parallel transport of a connection on a principal ∞-bundle over $X$: this assignment of k-morphisms in $A$ to $k$-dimensional paths in $X$ is the assignment of the classical action to trajectories in $X$.

More precisely, this is the classical action assigned to topologically disk-shaped $k$-paths. The full classical action of a classical σ-model quantum field theory is an extension of $\nabla$

from disk-shaped cobordisms to all cobordisms in $X$ (possibly and usually after an extension of the codomain to some new codomain $V$). This extension $\exp (i{S}_{\nabla })$ knows not just the parallel transport, but also the holonomy of the classical background field.

The question to be analyzed here is how this classical action $\exp (i{S}_{\nabla }):{\mathrm{Bord}}_n(X)\to F$ transmutes systematically to an FQFT $Z_{\nabla }:{\mathrm{Bord}}_n\to C$ on abstract bordisms (possibly with extra structure, if we pass from TQFT to richer theories like CFT). This step should be the path integral:

in some way the assignment of $Z_{\nabla }$ to a cobordism $\Sigma$ is expected to be obtained by “summing” in some way the value of $\exp (i{S}_{\nabla })$ on elements in the fiber of the forgetful functor ${\mathrm{Bord}}_n(X)\to {\mathrm{Bord}}_n$ over $\Sigma$, i.e. over all possible ways to map $\Sigma$ into $X$ – the sum of $\exp (S_{\nabla }(\varphi ))$ over all paths or field configurations $\varphi :\Sigma \to X$.

The basic idea that we shall follow has been sketched to some extent in the notes

• Nonabelian cocycles and their sigma model QFTs.

These notes in turn have grown out of the blog entry An Exercise in Groupoidification – the path integral from which the present entry draws its name. We hope to refine the discussion a bit more now, provide more details and more systematics.

General procedure

This section describes the general formalism that we want to study as a candidate for (aspects of) a fundamental abstract-nonsense description of quantization. The next section then picks special (simple) cases and works out the action of the abstract machinery on these.

the notion of space

We need to talk about spaces of sorts:

• a space $X$ that is the target space – often, but not always, to be interpreted as physical spacetime – in which the $d$-dimensional object – for $d=0$ for instance a point particle such as an electron – whose dynamics we want to encode propagates.

  This space may be just a manifold but we want to allow it to be a kind of space a bit more general than that, for instance an orbifold.

  When we are describing gauge theory the target space is smooth refinement of a classifying space $BG$.

  When we are describing finite gauge theory such as Dijkgraaf-Witten theory, the target space is just the one-object groupoid $BG$ that is the delooping of the gauge group $G$.

• for each $(d+1)$-dimensional manifold $\Sigma$ a path space or configuration space $[\Sigma ,X]$ of maps from $\Sigma$ into $X$.

The general abstract nonsence for dealing with general spaces of this sort is that of space and quantity. As described at motivation for sheaves, cohomology and higher stacks, this leads one to describe a space as an infinity-stack on some site $S$.

In the simplest case, which the reader should keep in mind, the site in question is the point and an infinity-stack on it is just an infinity-groupoid. In turn in simple cases of this, which the reader should still keep in mind, this infinity-groupoid will be just a groupoid or at most a 2-groupoid.

Whichever choice one makes, the collection of all such generalized spaces modeled on a certain site $S$ forms an (infinity,1)-topos $H$.

Despite possibly its appearance, the reader should take this statement as suggesting an immense simplification instead of a huge complification: the language of (infinity,1)-topos is a user interface that makes pretty sophisticated and rich notions of generalized spaces have the same look-and-feel as just plain topological spaces. It’s the inner workings of the formalism that will take of things coming out right even when we talk about richer objects.

In particular, it is the machinery of models for infinity-stack (infinity,1)-toposes that provides all the tools for actually performing the operations that we shall consider. The reader unfamiliar with that will suffer little loss from concentrating his or attention on the toy examples where all “spaces” involved are finite groupoids and trust that everything goes through analogously also in the other examples.

the notion of path

With an ((∞,1)-topos? context $H$-fixed we are in a context that allows us to study cohomology. A classical background field on a target space $X$ is in parts a cocycle on $X$ with values in some coefficient object $A\in H$.

But, as indicated above, it is actually, more: it is a differential cocycle on $X$: something that depends not just on $X$ itself, but also on a notion of paths in $X$.

To encode this we may pick a cosimplicial object

in our $(\mathrm{\infty }\mathrm{,1})$-category of spaces, that encodes which object in $H$ models the standard $k$-simplex regarded as a space. For instance if $H$ is the (infinity,1)-topos of Lie ∞-groupoids modeled by ∞-stacks on Diff, we would take ${\Delta }_H^n$ to be the standard $n$-simplex ${\Delta }_H^n\subset {ℝ}^n$ regarded as a smooth manifold in the standard way.

Then the space of disk-shaped $k$-dimensional paths in $X$ is $[{\Delta }_H^n,X]$.

These spaces glue together to the path infinity-groupoid

This is an ∞-groupoid whose k-morphism are generated from the original $k$-morphisms of $X$ and the $k$-dimensional path $\varphi :{\Delta }_H^k\to X$.

For instance if $X=Y//G$ is a global orbifold, the 1-morphisms in $\Pi (X)$ will be generated from smooth paths $\gamma :y_1\to y_2$ in $Y$ as well as orbifold jumps $g:y_1\to g(y_1)$ subject to the relation

Or if $X=C(Y)$ is realized as the Cech nerve of some Cech cover $(U={\coprod }_i{U}_i)\to Y$ then $\Pi (X)$ will be generated from paths $(\gamma ,i):(y_1,i)\to (y_2,i)$ in $U_i$ for all $i$ and transitions $(y,i)\to (y,j)$ for all $y\in U_i\cap U_j$ subject to the condition

the notion of trajectory

More generally, let $D$ be some poset and specify a functor

We think of as the image of some $k\in D$ under ${\Sigma }_H$ as the realization in $H$ of some abstract cobordism whose boundary component inclusions are the images ${\Sigma }_H^j\to {\Sigma }_H^k$ of all morphisms $j\to k$ in $D$.

So ${\Sigma }_H$ encodes a multi-cospan of cobordisms in $H$. In principle ${\Sigma }_H$ will in general be supposed to range over all cobordisms in some sense, but for our discussion it will be complete sufficient to concentrate on much less. Most of our discussion concerns a handful of cobordisms, usually just two of them and their composites.

In any case, by slight abuse of notation, we shall write

and speak of the $\mathrm{\infty }$-groupoid of bordisms in $X$. If we assume that ${\Sigma }_H$ ranges at least over the disk-shaped cobordisms ${\Delta }_H^k$ we have a canonical inclusion

the notion of background field

a flat classical background field (with vanishing field strength) is a morphism

(the more general situation is described at …, but need not concern us here for the moment. It’s quantization will structurally be a bit, but not essentially different from that of the flat case)

this encodes an $A$-principal ∞-bundle with connection.

In the case that $A=BG$ is the delooping of an $\mathrm{\infty }$-group $G$, so that $A$ has a unique point ${\mathrm{pt}}_A:*\to BG$ the total space of this principal ∞-bundle is the (homotopy) pullback $P$ in

where $\nabla {\mid }_{\mathrm{pt}}$ is the component of $\nabla$ (as a morphism out of a colimit) on points in $X$.

We are interested in forming the associated $\mathrm{\infty }$-bundle induced by a choice of representation

on some (∞,1)-category $V$. As a cocycle this is just the composite

This is the background field that encodes the forces acting on the objects propagating in $X$ whose quantum dynamics we seek.

We will often abuse notation and write just $\nabla$ for ${\rho }_{*}\nabla$, when the context is clear.

the notion of action functional

A choice of extension of this through the inclusion $\Pi (X)\to \mathrm{Bord}(X)$ is the corresponding action functional $\exp (S_{\nabla })$

For each cobordism $\Sigma$ the component $\exp (S_{\nabla }){\mid }_{\Sigma }$ of the action functional will define a total space of an associated bundle $E_{\Sigma }$. These are discussed in the next section.

the notion of section and state

A state of the quantum theory over a cobordism $\Sigma$ is supposed to be a section of the bundle $E_{\Sigma }\to [\Sigma ,X]$ over the configuration space $[\Sigma ,X]$ of fields on $\Sigma$, where we allow generalized sections that may be distributional.

In order to define sections and states, we need to fix a point of the coefficient object $V$, i.e. a morphism

from the terminal object $*$ into $V$. The choice of this point determines over which monoid our quantum theory will be linear:

definition (ground monoid)

given the pointed object $*\stackrel{{\mathrm{pt}}_V}{\to }V$ we say that the respective ground monoid is the endomorphism monoid

i.e. the lax pullback

example: ordinary linear quantum mechanism

In the special case that $V=$ Vect over some ground field $k$ the canonical point is that ground field and then the “ground monoid” is again canonically identified with the ground field: ${\mathrm{End}}_{{\mathrm{Vect}}_k}(k)\simeq k$. Therefore the name “ground monoid” as a generalization of this situation.

More generally, for $W:*\to V$ any other point of $V$, we say that the lax pullback

is the object of elements of $W$ relative to the chosen point.

example: ordinary vectors

Again in the special case that $V=$ Vect with the above standard point, $W\in V$ will be some vector space and ${\mathrm{el}}_k(W)$ is the set, regarded as a discrete category of its ordinaly elements – the vectors in $W$.

Moreover, given the background field differential cocycle or the action functional

we have the lax pullback

Next we make use of the fact that we are in fact in an (infinity,1)-topos so that colimits are pullback stable… (not yet, really)

For each component $\exp (S_{\nabla }){\mid }_{\Sigma }:[\Sigma ,X]\to V$ we get a corresponding lax pullback

in particular the fiber over $x\in X$ of $E_{\mathrm{pt}}$

is the fiber of the underlying $V$-$\mathrm{\infty }$-bundle

example

in the case that $V=\mathrm{FinVect}$ and $\nabla :\Pi (X)\to \mathrm{Vect}$ is a vector bundle with connection and with the canonical point chosen for $\mathrm{Vect}$ as above, this ${\mathrm{el}}_{{\mathrm{pt}}_V}(E_x)$ is simply the fiber of that vector bundle over $x$.

Choose the geometric function theory given by over categories in $H$ as described at examples for geometric function objects

a section of the background field or state of the quantum system is over each point $x$ an object $C(E_x)$.

special case: degroupoidification for $V$-phased groupoids

In the case that $V=\mathrm{FinVect}$ we have that $E_x$ is (the discrete category on the set underlying) a vector space.

For $(\psi :\Psi \to E_x)\in C(E_x)$ consider the underlying morphism of plain $\mathrm{\infty }$-groupoids and interpret this as the vector

given by the linear combination of all elements $v\in E_x$ weighted by the groupoid cardinality of the infinity-groupoid sitting over it.

the propagation map

In total the above discussion yields, starting from a background field $\nabla :\Pi (X)\to V$, a multispan of $\mathrm{\infty }$-bundles $E_{\Sigma }$ over configuration spaces $[\Sigma ,X]$.

An an object in the over category $C(E_{\Sigma }):=H/E_{\Sigma }$ encodes a – possibly distributional – section of $E_{\Sigma }\to [\Sigma ,X]$. The quantum propagation we take to be the pull-push of the geometric function objects $C(E_{\Sigma })$ through this multispan.

Let ${\Sigma }_{\mathrm{in}}\to \Sigma \leftarrow {\Sigma }_{\mathrm{out}}$ be a cobordism cospan regarded as part of the multi-cospan ${\Sigma }_H$ that defines $\mathrm{Bord}(X)$ as

this realizes $\mathrm{Bord}(X)$ as the universal cocone whose component under $\Sigma$ is

by lax pullback of ${\mathrm{pt}}_V:*\to V$ this induces a span of total spaces

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definition the “quantum propagation along $\Sigma$” is the functor of geometric function objects

given $\psi :\Psi \to E_{\mathrm{in}}$ we extract the corresponding “degroupoidified” vector of the corresponding section over $\varphi {\mid }_{{\Sigma }_{\mathrm{in}}}\in [{\Sigma }_{\mathrm{in}},X]$ as the pullback

Concrete examples

electromagnetically charged particle

..

take $A=BU(1)$ such that

encodes the electromagnetic field

and choose the standard representation

then the above propagation prescription sends the $\delta$-section ${\delta }_x$ \to $U{\delta }_x:y\mapsto$ = sum over all paths $\gamma :x\to y$ of $\exp (S_{\nabla }(\gamma ))$.

Dijkgraaf-Witten theory

Let $X=BG$ for $G$ a finite group. Then $[\Sigma ,BG]=G\mathrm{Bund}(\Sigma )$ is the orbifold of $G$-principal bundles on $\Sigma$.

Consider for simplicity just codimension 1. To do that regard $\mathrm{Bord}(BG)$ as a 1-category whose objects are 2-dimensional closed manifolds ${\Sigma }_{\mathrm{in}},{\Sigma }_{\mathrm{out}}$ and whose morphism corresponding spans of 3-dimensional cobordisms ${\Sigma }_{\mathrm{in}}\to \Sigma \leftarrow {\Sigma }_{\mathrm{out}}$.

Then

is under the above “degroupoidification” the action of Dijkgraaf-Witten theory, where the $\frac{1}{\mid \mathrm{Aut}(P)\mid }$-weight comes from the fact that the pulled back state ${\mathrm{in}}^{*}\psi$ picks up the automorphisms of $P$ by the push-forward

the Yetter model

Proposition.

The combinatorial weighting for the “categorified DW” theory introduced by Yetter (and studied by Girelli, Pfeiffer, Popescu, Porter, Martins etc.) is indeed the Leinster measure of the corresponding configuration space.

Recall that for ordinary DW theory for finite group $G$ config space is the groupoid $[\Sigma ,BG]$, which, since $\Sigma$ is finite, may be computed by choosing a triangulation of $\Sigma$, letting $P_1(\Sigma )$ denote the free category on the corresponding graph and considering $\mathrm{Func}(P_1(\Sigma ),BG)$.

Since there are

many ways to assign group element to the $\nu$ vertices of the triangulation, there are $\mid G{\mid }^{\nu }$ many natural transformations starting at any functor $P_1(\Sigma )\to BG$ and therefore the Leinster measure of the hom-groupoid is

Now, Yetter et al find that when target space is replaced by

for $G_{(2)}$ the finite strict 2-group coming from the strict crossed module

then the corresponding combinatorial weight becomes

where $\nu$ is the number of vertices, as before, and where $\lambda$ is the number of edges in $X$.

(See for instance theorem 2.13 on p. 127 of Martins-Porter’s On Yetter's invariants or the discussion below theorem III.2 on p. 6 of Girelli-Pfeiffer-Popescu’s BF to BFCG (while the factor is not manifest in their expression (15))).

The claim here is that this factor is precisely the Leinster measure on the configuration space

Here I am taking, to replace awkward pseudofunctors with nicer strict 2-functors, $P_2(\Sigma )$ now to be the strict 2-groupoid generated from the triangles in $\Sigma$.

Then the above hom-groupoid is supposed to be the 1-groupoid obtained by passing to equivalence classes of 1-morphisms in the 2-groupoid whose objects are strict 2-functors $P_2(\Sigma )\to B{G}_{(2)}$, whose morphisms are pseudonatural transformations of these, and whose 2-morphisms are modifications of those.

To prove this, we need to compute how many 1-morphisms in $\mathrm{conf}$ there start at any given object.

But that’s easy: a 1-morphism

in the 2-groupoid of strict 2-functors $P_2(\Sigma )\to B{G}_{(2)}$ is a transformation whose component is an assignment of squares in $B{G}_{(2)}$ to edges in $X$:

By the general yoga of strict 2-groups, we know that the square on the right is entirely and precisely fixed by

• the source 2-functor value $A(\gamma )\in G$ on the top;

• the two transformation elements $g(x),g(y)\in G$ on the two sides;

• and the label $g(\gamma )\in H$ of the 2-morphism filling the square.

This uniquely determines the value $A\prime (\gamma )$ of the pseudofunctor which is the target of the transformation whose component $g$ is.

Therefore, clearly, there are precisely

many 1-morphisms in

starting at any given object.

But some of them are related by 2-morphisms

To figure out many, we need to compute how many modifications $\eta :g\to g\prime$ there are starting at any transformation $g$.

But again the same logic applies: such a transformations is given by a component map which sends vertices in $X$ to 2-cells in $B{G}_{(2)}$:

for $\eta (x)\in H$.

Again one sees, by drawing the relevant naturality tin-can diagram which I won’t bother to do here in MathML, that for each such choice of assignments of elements in $H$ to vertices in $X$ there is precisely one modification $\eta :g\to g\prime$.

This means that we have

• $\mid G{\mid }^{\nu }\mid H{\mid }^{\lambda }$ 1-morphisms emanating at each object

and

• $\mid H{\mid }^{\nu }$ 2-morphisms emanating at every morphism.

The Leinster measure in this situation is

precisely the combinatorial factor which people found makes the Yetter state sum model a topological invariant.